Let z be a complex number satisfying `|z-5i|<=1` such that `amp(z)` is minimum, then z is equal to

Options

A. `(2sqrt(6))/(5)+(24i)/5`
B. `24/5+(2sqrt(6)i)/(5)`
C. `(2sqrt(6))/(5)-(24i)/5`
D. none of these

Answer

To solve the problem, we need to find a complex number \( z \) that satisfies the condition \( |z - 5i| \leq 1 \) and has the minimum amplitude (or argument).

### Step-by-step Solution:

1. **Understanding the Condition**:
The condition \( |z - 5i| \leq 1 \) describes a circle in the complex plane centered at \( 5i \) (which is the point \( (0, 5) \) in the Cartesian plane) with a radius of 1.

2. **Expressing \( z \)**:
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. The condition can be rewritten as:
\[ |(x + (y - 5)i)| \leq 1 \]
This implies:
\[ \sqrt{x^2 + (y - 5)^2} \leq 1 \]

3. **Finding the Center**:
The center of the circle is at \( (0, 5) \), and the radius is 1. Therefore, the points that satisfy this condition are within the circle defined by:
\[ (x - 0)^2 + (y - 5)^2 \leq 1 \]

4. **Minimizing the Amplitude**:
The amplitude (or argument) of \( z \) is given by \( \tan^{-1}(\frac{y}{x}) \). To minimize this, we want to find the point on the boundary of the circle that has the smallest angle with the positive x-axis. This occurs when \( y \) is minimized while \( x \) is as close to 0 as possible.

5. **Finding the Lowest Point on the Circle**:
The lowest point on the circle occurs when \( y = 4 \) (since the center is at \( y = 5 \) and the radius is 1). Thus, we have:
\[ y = 4 \]
For \( x \), the closest point to the y-axis (where \( x = 0 \)) would be \( x = 0 \).

6. **Conclusion**:
Therefore, the complex number \( z \) that satisfies the given conditions is:
\[ z = 0 + 4i = 4i \]

### Final Answer:
\[ z = 4i \]